The Radius Of The Earth

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2017/07/12 - The Radius of the Earth

The talk/workshop I give about computing the distance to the Moon uses, it claims, nothing more than a pendulum and a stopwatch. And while it's sort of true that it uses nothing else, it's not really true, because it also uses the period of the Moon, and the size of the Earth.

Now it might be possible to persuade you that it's OK to use the period of the Moon, since you can simply look out the window and measure that for yourself, but to use the size of the Earth seems a bit of a stretch. Surely there's no way to compute that from your back garden.

Of course, we could claim that since the original definition of the metre was:

"One 10 millionth of the distance from
the North Pole to the Equator on the
longitude that runs through Paris"

then perhaps we can quite reasonably claim that the Earth's circumference is, by definition, 40 million metres, but even so, perhaps that's not really fair, not really playing the game.

But when I showed Distance to the Moon calculation to a friend of mine, it turned out he (a) was really interested, and (b) suggested a method to compute the radius of the Earth from your back garden.

If you don't know the Eratosthenes method you really need to go look it up. It's beautifully simple and elegant.
Mike, as it would happen, is an expert on sundials, so he really knows his stuff, and while we all (admittedly for some definition of "all") know Eratosthenes' method for measuring the size of the Earth using shadows of sticks, reflections in wells, and distances between cities, Mike told me about a completely different technique.

It goes like this ...

Stand on a wall, perhaps two metres high, and wait for sunrise. When you see the sun just peak above the horizon, start the stopwatch, and jump off the wall (or drop the stopwatch to a friend). Since you are now lower down, you can no longer see the Sun, so wait a bit until you can see it again, and stop the clock.

And astonishingly, that's enough!

OK, so in truth you need to be able to see the horizon, so it actually works best at the beach, but even so, in theory, and even in practice, this is enough to calculate the size of the Earth.

So your challenge is this:

  • How does it work?
  • What do you need to do to correct for latitude?
  • How accurate is the technique?
  • For a 2m wall, how long will the timing be?
  • Can you improve the experimental setup?

We can then ask similar questions, such as:

  • What is our latitude?
  • How is this affected by the axial tilt?
  • What is the axial tilt?

And so the initial measurements can be refined to give a more accurate answer. As always, knowing the error bars will be important.

This could be an absolutely brilliant project for a suitably able and motivated student.

Or you.

If you want to see the calculation, it's on this page:

My thanks to several comments, including some from Rob Low via Twitter and email.

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